Vlasov solvers, solving directly the Vlasov equation on a grid of phase-space, have been successful in the past years in solving problems as well in plasma physics as in beam physics that were inaccessible to Particle-In-Cell (PIC) methods because of their inherent large numerical noise. However, in simulations where the particle distribution function varies a lot during time, Vlasov simulations on a uniform grid of phase space become very inefficient because large computational resources are waisted on regions where nothing is happening at a given time. For this reason we are investigating methods using a phasespace grid which evolves in time according to the evolution of the distribution function. Two different paths have been taken. The first consists in having a uniform grid at each time step, but which is moving from one time step to the next according to the global evolution of the particles. The second approach, which eventually could be coupled to the first, consists in using an adaptive local mesh refinement scheme. The adaptive method is overlayed to a classical semi-Lagrangian method which is based on the conservation of the distribution function along particle trajectories. The phase-space grid is updated using a multiresolution technique. In this presentation we shall describe the moving grid and adaptive mesh refinement methods and evaluate their benefits on some applications in beam physics and plasma physics.